A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

(v) The system is closed so that neither bacteria nor substrate enter (exit) through
the boundaries.
We can write the system of equations 8 (in the same fashion as done in section 2.1)
as
∂b
∂t
∂s
∂t
�
∂
= µ(s)
∂x
∂b
�
∂s
− bχ(s)
∂x ∂x
17
(2.10)
= −k(s)b (2.11)
for t > 0 and 0 < x < L (L=length of capillary tube), and with the conditions
s(x, 0) = s0(x), b(x, 0) = b0(x),
∂b
∂x
= ∂s
∂x
= 0 when x = 0, L.
The no flux conditions are because of assumption (v), and s0 and b0 are given initial
distributions of substrate and bacteria, respectively. Since the focus of this study is
to determine if traveling waves can be observed, and since the length of the capillary
tube is considered to be very large in comparison to a bacterial band, the analysis
is facilitated by introducing a traveling wave coordinate and extending the domain
of interest to the whole real line. That is, let
z = x − ct − ∞ < z < ∞, b(x, t) = b(z), and s(x, t) = s(z)
Further assume that k, µ, and s0 are constant and that in the far field s = s0. In
the variable z the system (2.10)–(2.11) becomes 9
cb ′ = −µb ′′ + (bχ(s)s ′ ) ′
(2.12)
cs ′ = −kb (2.13)
8 Observe that in this one space dimension setting the operator ∇ = ∂
∂x .
9 ′ ≡ d
dz